Climate blog-viating being what it is, readers interested in whether the current flat trend in data is consistent with 2C/century are reading terms like “ARMA(1,1)” ; those who have been reading my blog are aware that I have extended my hypothesis tests to include results we obtain if we use a statistical model called “AR(1)+ White Noise”.

Some of you may be wondering, “Does the choice between these two models make much of a difference when testing the 2C/century trend hypothesis?” The answer is: Practically none.

Interestingly enough, when “AR(1) + White Noise” can be used to describe a process, ARMA(1,1) can also be used. That is to say, in this circumstance: The models exactly the same. (The equations look different though. That’s math for you. 🙂 )

So, why are we reading about both processes at blog? Including this one?

Well, I prefer to use the “AR(1)+White Noise” idea because I like to describe models in a way that give me some chance of distinguishing artifacts due to measurement errors from phenomenology.

Since the “white noise” at least might be related to measurement uncertainty, this model lets me compare the magnitude of “white noise” in the data process and the magnitude estimated by measurement agencies like Hadley. In contrast, ARMA(1,1) makes provides nothing which can be compared to measurement uncertainty. Instead, it provides a parameter that describes the effect of the “moving average” of white noise driving the process. Maybe someone somewhere can relate this individual process to something physical about the climate or something about measurement uncertainty. I can’t see any obvious connection.

Describing the “noise” in measured weather as “AR(1) + White Noise” seems better suited than ARMA(1,1) to this purpose. But some other people prefer to describe the measurements of weather noise using ARMA(1,1), so both must be discussed.

The remainder of this blog post will cover two topics

1. More detailed background on the motivation for describing “noise” in observations of GMST using an “AR(1)+White Noise process.”
2. Show the correlograms of the two processes to demonstrate they really are the same.

Background on AR(1)+White Noise.

Readers with backgrounds in the sciences and engineering are generally familiar with “white noise” for a number of reasons One of these is that measurements of physical quantities like, for example Global Mean Surface Temperature (GMST), often contain errors. When each error in sequence of measurements is independent of errors all other measurements, and the magnitude of errors is thought to be identical over the set of measurements, these measurement errors are often modeled as “white noise”; in addition, the distribution of errors is often assumed to be “Gaussian”.

In short: “Gaussian white noise” is an extremely common assumption for measurement errors. One generally assumes this form for noise unless there is identifiable reason to suggest it does not apply.

So, what about AR(1)?

Scientists and engineers are also often familiar with “AR(1)” noise, also called “red” noise. Red noise also manifests itself in many physical processes of interest. For example, in Brownian motion, very small particles suspended in a fluid are knocked around by the random motion of individual molecules of viscous fluid. The result is the particles diffuse. It turns out that if we assume the “kicks” by the molecules obeys something called “the Weiner process”– which is the derivative of a “Gaussian white noise” process, if we sample the velocities of the particles at equal time intervals, that process will be described by an AR(1) process. (See Hoel, Port, Stone, chapter 4.3).)

The AR(1) processed also rocketed to the attention of climate blog addicts when Stephen Schwartz published his paper discussing the time constant of the earth’s climate. His empirical estimate of the climate time constant was based on the assumption that “weather noise” for the earth could be described using an AR(1) process.

I discussed that paper in Dec. 2007. In that post, I agreed with Tamino that the residuals to an ordinary least squares (OLS) fit for observations of global means surface temperature (GMST) were imperfectly described by an AR(1) process. However, I also observed the individual measurements of GMST are known to include measurement uncertainty. If we assume that some of that measurement uncertainty is “white noise”, then we would necessarily also conclude that the process describing measurements of GMST would be “AR(1) + White Noise.”

It turns out this model fits the earth data pretty darn well. Moreover, if we assume the noise in measurements consists of AR(1) “weather noise” and white measurement noise, the magnitude of “weather noise” is in fair agreement with estimates from Hadley.

This doesn’t prove “AR(1)+White Noise” is the correct description, nor does it prove the AR(1) portion is the “weather part” and the “white” portion is due to measurements. However, the idea that measurements of “weather noise” is described by “AR(1)+White Noise” works fairly well, which suggest Schwartz’s model was “not inconsistent with” the measurements of GMST over hundreds of years. (This is not the same as saying the full method works, as there are other difficulties, including issue associated with de-trending during a period when the climate trend is thought to vary.)

Regardless of the accuracy of Schwartz’s final result, when the “AR(1)+White Noise” statistical model has these attractive features:

1. AR(1) noise has been an acceptable modeling assumption for “weather noise”, appearing in peer reviewed literature. This includes not only Schwartz’s paper, but papers he cited.
2. White noise is the most common assumption used to describe measurement errors in time series. It is plausible that at least some portion of measurement errors in time series are white.
3. The correlogram for earth’s temperature fits this model fairly well.
4. Under this modeling assumption, it is possible to obtain separate estimates of the amount of “white” and “AR(1)” noise. This permits us to compare the estimate of the “white” noise to estimate of the measurement noise from the measurement groups. (Hadley provides estimates of uncertainty on their graphs and data here.)

What about ARMA(1,1)?

More recently, Tamino suggested ARMA(1,1). So, is this model “in competition” with “AR(1) + White Noise”. Nope. As Tamino observes graciously in comments after his recent post:

I suspect she’s not aware that the two models are equivalent in the sense that every model which is the sum of an AR(1) process and a white-noise process can be reformulated as an ARMA(1,1) process. However, not every ARMA(1,1) model can be reformulated as the sum of an AR(1) process and a white-noise process.

In other words: When a process is AR(1) + White Noise, it is always also ARMA(1,1). Yes, I was aware of this.

As for the other bit: Tamino is telling his readers that there are ARMA(1,1) processes that cannot be described as AR(1)+White Noise. Yep. I was aware of that too! 🙂

In case you aren’t convinced by either Tamino or my statements that the when AR(1)+White Noise describes data, it is also ARMA(1,1), I’ll show you the correlograms. You’ll see both in my post on Schwartz’s articles and Tamino’s that the correlation at lag “n” for the two process obeys ρ_n=C φⁿ, where C is some constant that depends on the parameters of the process and φ is the parameter for the AR(1) portion of either process. This means if we plot the natural log of the correlation as a function of “n”, the two processes will fall on the same curve. But that’s just an equation, right?

Graphs are better!

To make graphs, I fit the parameters that best fit the HadCrut 3 observations from 1975-now. This is the period Tamino chose to estimate the process in his post. It contains numerous strong volcanic eruptions.

For ARMA(1,1), the “noise” in the observations of GMST is modeled using:
1x_n=w_n+ φ x_n-1+θ w_n-1

I obtained φ=0.8396, θ=-0.4046 and σ_w=0.1153.

For “AR(1)+White noise” the process equation is described in two parts:

2ax_n=w_n+ y_n
with
2ay_n=φ y_n-1+v_n

Using a method similar to that described yesterday, but adapted to the new process, I obtained:
φ=0.8396, σ_v=0.0674 and σ_w=0.0800.

Once I had these parameters, I ran some twenty simulations of 400 months weather for each process. I then averaged the results over the twenty simulations plotted the average of the correlograms. Voila!

Do those two correlograms look the same or what? In case you’d like to see plots of the natural log of the correlation as a function of lag, here it is:

As you can see, the average of the correlogram is identical for both processes. So: Theory says the two processes are the same. If we fit these to identical sets of data, they come out the same. Yes, they are the same. Fancy that?

Does mean we’ll always get precisely the same number regardless of choice of processes: No. We will get nearly identical values. However, while I posted parameters to four significant figures above, there is uncertainty in those parameters. Also, because the methods of estimating the parameters use slightly different equations. When the parameter are estimated, some rounding will take place. So, there will be some slight differences in numerical results.

This means we will get slightly different distributions of probabilities when we run monte-carlo using one set of parameters rather than another. However, the choice of ARMA(1,1) vs. ‘AR(1)+Noise’ is not one that will generally make much difference in the outcome of a hypothesis test of 2C/century.

Wrap up

Going forward, as you read some are using ARMA(1,1) for hypothesis tests and other use ‘AR(1)+Noise’, remember: This choice has little practical importance.

In future posts, I’ll be discussing the analysis decision that does have practical importance. This is the choice of time period used to estimate the parameters in used in the equations. If the time period exhibits large temperature variations due to volcanic eruptions like El Chicon and Pinatubo, then those swings will result in parameters that create simulated weather with similarly large temperature swings. If the cause of the temperature swings was the eruptions of El Chicon and Pinatubo, then it’s plausible the variability of weather when during calmer periods will be over-estimated as a result of an inappropriate analytical assumption.

So, while this post is long, (and oddly necessary,) it only serves to show the choice of ARMA(1,1) vs. AR(1)+White noise is relatively unimportant with regard to the issue of testing whether 2C/century is consistent with weather.

Addressing that unimportant question will let us move on to the more important question, which is is:
“Can we estimate the distribution of 7 year (or 8, 9, 10 etc. year ) trends during distribution of temperature swings during periods when there are no eruptions of the magnitude of Pinatubo and El Chicon using observations that include those eruptions?

I don’t think the answer is “no”. Or, at best, the answer is, “One has a legitimate right to doubt estimates of variability obtained that what.”

Later on, we’ll see if I can convince others. 🙂